def test():
A = Uged(a, t1 + t2) * Ugde(d, t3) * Uegd(a, t2)
print(
evaluate_cumulant(A,
positive_times=(t1, t2, t3),
leading_index=a,
arrays=["gg"]))
def print_trans_R2g_alt():
"""
"""
#A = (Uedg(a,t1+tau)*Ugde(b,t1+tau)*Uedg(b,t1+t2)*Ugde(b,t1+t2+t3)
# *Uedg(b,t1+tau)*Ugde(a,t1+tau)*Uedg(a,t1))
A = (Uged(a, t1) * Uedg(a, tau1) * Ugde(b, tau1) * Uedg(b, t2) *
Ugde(b, t2 + t3) * Uedg(b, tau1) * Ugde(a, tau1))
print(
evaluate_cumulant(A,
positive_times=(t1, t2, t3),
leading_index=a,
arrays=["gg"]))
def generate_nth_order_R2g(states_tuple, times_tuple):
order = len(states_tuple)
if order != len(times_tuple):
raise Exception("Wrong tuple/list length")
# starting state
a = states_tuple[0]
# final state (can be the same as starting)
b = states_tuple[len(states_tuple) - 1]
# final time (must be t2)
tt = times_tuple[len(times_tuple) - 1]
AL = Uged(a, t1)
Amid = Uedg(b, tt) * Ugde(b, t3 + tt)
filL = 1
filR = 1
for k in range(len(times_tuple) - 1):
tau = times_tuple[k]
s1 = states_tuple[k]
s2 = states_tuple[k + 1]
filL = filL * Uedg(s1, tau) * Ugde(s2, tau)
filR = Uedg(s2, tau) * Ugde(s1, tau) * filR
A = AL * filL * Amid * filR
print(A)
print(
evaluate_cumulant(A,
positive_times=(t1, tt, t3),
leading_index=a,
arrays=["gg"]))
<a|H(t)|b><c|W|d>
= J_ac*<\Psi_d|Dagger(U_a(t))*U_b(t)|\Psi_c>/Norm
= J_ac*<\Psi_g|U_g(T)Dagger(U_d(T))Dagger(U_a(t))U_g(t)
x Dagger(U_g(t))U_b(t)U_c(T)Dagger(U_g(T))|\Psi_g>/Norm
= J_ac*<\Psi_g|[U_g(T)Dagger(U_d(T))][Dagger(U_a(t))U_g(t)]
x [Dagger(U_g(t))U_b(t)][U_c(T)Dagger(U_g(T))]|\Psi_g>
x Norm^-1
=> Uged(d,T)*Uedg(a,t)*Ugde(b,t)*Uegd(c,T)*(1/Norm)
Norm = Uged(d,T)*Uegd(c,T)
"""
A = Uged(d,T) *Uedg(a,t)*Ugde(b,t)* Uegd(c,T)
Anorm = Uged(d,T)*Uegd(c,T)
verbatim = False
if verbatim:
print(" ")
print("Expression to evaluate: ")
print(" ")
print("Tr_bath{",A,"W_eq}")
print(" ")
print(" ")
A = A.rewrite(gg)
n = Symbol('n')
t1 = Symbol('t1')
k = Symbol('k')
k1 = Symbol('k1')
#A = Uged(n,T)*Uedg(n,tau)*ExpdV(a,tau,x)*Ugde(m,tau)*ExpdV(b,0,y)*Uegd(n,T)
#A = Uged(b,T)*ExpdV(n,0,x)*Uegd(a,tau)*ExpdV(m,-tau,y)*Uged(b,tau)*Uegd(b,T)
#A = Uedg(b,t)*ExpdV(m,0,x)*Ugde(a,t)*Uedg(a,tau)*ExpdV(n,0,y)*Ugde(b,tau)
A = Uedg(b,t)*ExpdV(m,t,x)*Ugde(a,t)*Uedg(a,tau)*ExpdV(n,tau,y)*Ugde(b,tau)
#A = Uedg(b,t)* Ugde(a,t-tau)* Ugde(b,tau)
#A = Uedg(b,tau)*ExpdV(m,tau,x)*Ugde(a,tau)*Uedg(a,0)*ExpdV(n,0,y)*Ugde(a,0)
#A = Uedg(b,t)*ExpdV(m,t,x)*Ugde(a,t)*Uedg(a,tau)*ExpdV(n,tau,y)*Ugde(a,tau)
Anorm = Uged(n,T)*Uegd(n,T)
verbatim = True
if verbatim:
print(" ")
print("Expression to evaluate: ")
print(" ")
print(" Tr_bath{",A,"W_eq}")
print(" ")
print("The expression is normalized by:")
print(" ")
print(" Tr_bath{",Anorm,"W_eq}")
print(" ")
#A = Uedg(b,t)*ExpdV(m,0,x)*Ugde(a,t)*Uedg(a,tau)*ExpdV(n,0,y)*Ugde(b,tau)
#A = Uedg(b,t)*ExpdV(m,t,x)*Ugde(a,t)*Uedg(a,tau)*ExpdV(n,tau,y)*Ugde(b,tau)
#A = Uedg(b,t)* Ugde(a,t-tau)* Ugde(b,tau)
#A = Uedg(b,tau)*ExpdV(m,tau,x)*Ugde(a,tau)*Uedg(a,0)*ExpdV(n,0,y)*Ugde(a,0)
#A = Uedg(b,t)*ExpdV(m,t,x)*Ugde(a,t)*Uedg(a,tau)*ExpdV(n,tau,y)*Ugde(a,tau)
#A = Uged(e,T)*Uedg(a,t)*Ugde(b,t)*Uedg(c,t-tau)*Ugde(d,t-tau)*Uegd(d,T)
""" Parts of the relaxation tensor """
part = "test3"
""" M1 K """
if part == "M1_K":
A = Uged(b,T)*Uedg(b,t)*ExpdV(m,t,x)*Ugde(a,t)*Uedg(a,t-tau)\
*ExpdV(n,t-tau,y)*Ugde(b,t-tau)*Uegd(b,T)
use_norm = False
Anorm = Uged(n,T)*Uegd(n,T)
nderiv = 2
filename = "m1_K.txt"
elif part == "M1_H":
pass
elif part == "K_bk":
= <\Psi_g|Dagger(U_e(T))Dagger(U_a(t))<a|dV|b>U_b(t)Dagger(U_c(t-tau))
<c|dV|d>U_d(t-tau)U_d(T)|\Psi_g>
= <\Psi_g|[U_g(T)Dagger(U_e(T))][Dagger(U_a(t))U_g(t)][Dagger(U_g(t)U_b(t)]
x [Dagger(U_c(t-tau))U_g(t-tau)][Dagger(U_g(t-tau)U_d(t-tau)]
x [U_d(T)Dagger(U_g(T))]|\Psi_g>
= Uged(e,T)*Uedg(a,t)*Ugde(b,t)*Uedg(c,t-tau)*Ugde(d,t-tau)*Uegd(d,T)
"""
m = Symbol('m')
n = Symbol('n')
t1 = Symbol('t1')
A = Uged(n, T) * Uedg(n, tau) * ExpdV(a, tau, x) * Ugde(m, tau) * ExpdV(
b, 0, y) * Uegd(n, T)
A = Uged(b, T) * ExpdV(n, 0, x) * Uegd(a, tau) * ExpdV(m, -tau, y) * Uged(
b, tau) * Uegd(b, T)
Anorm = Uged(n, T) * Uegd(n, T)
verbatim = True
if verbatim:
print(" ")
print("Expression to evaluate: ")
print(" ")
print(" Tr_bath{", A, "W_eq}")
print(" ")
print("The expression is normalized by:")
print(" ")
<a|H(t)|b><c|H(t-tau)|d><d|W|e>
= <\Psi_e|<a|H(t)|b><c|H(t-tau)|d>|\Psi_d>
= <\Psi_g|Dagger(U_e(T))Dagger(U_a(t))<a|dV|b>U_b(t)Dagger(U_c(t-tau))
<c|dV|d>U_d(t-tau)U_d(T)|\Psi_g>
= <\Psi_g|[U_g(T)Dagger(U_e(T))][Dagger(U_a(t))U_g(t)][Dagger(U_g(t)U_b(t)]
x [Dagger(U_c(t-tau))U_g(t-tau)][Dagger(U_g(t-tau)U_d(t-tau)]
x [U_d(T)Dagger(U_g(T))]|\Psi_g>
= Uged(e,T)*Uedg(a,t)*Ugde(b,t)*Uedg(c,t-tau)*Ugde(d,t-tau)*Uegd(d,T)
"""
A = Uged(e,T)*ExpdV(a,t,x)*ExpdV(b,t-tau,y)*Uegd(d,tau)
Anorm = Uged(e,T)*Uegd(d,T)
verbatim = True
if verbatim:
print(" ")
print("Expression to evaluate: ")
print(" ")
print(" Tr_bath{",A,"W_eq}")
print(" ")
print("The expression is normalized by:")
print(" ")
print(" Tr_bath{",Anorm,"W_eq}")
print(" ")
= <\Psi_g|Dagger(U_e(T))Dagger(U_a(t))<a|dV|b>U_b(t)Dagger(U_c(t-tau))
<c|dV|d>U_d(t-tau)U_d(T)|\Psi_g>
= <\Psi_g|[U_g(T)Dagger(U_e(T))][Dagger(U_a(t))U_g(t)][Dagger(U_g(t)U_b(t)]
x [Dagger(U_c(t-tau))U_g(t-tau)][Dagger(U_g(t-tau)U_d(t-tau)]
x [U_d(T)Dagger(U_g(T))]|\Psi_g>
= Uged(e,T)*Uedg(a,t)*Ugde(b,t)*Uedg(c,t-tau)*Ugde(d,t-tau)*Uegd(d,T)
"""
m = Symbol('m')
n = Symbol('n')
t1 = Symbol('t1')
A = Uged(n, T) * Uedg(n, tau) * ExpdV(a, tau, x) * Ugde(m, tau) * ExpdV(
b, 0, y) * Uegd(n, T)
Anorm = Uged(n, T) * Uegd(n, T)
verbatim = True
if verbatim:
print(" ")
print("Expression to evaluate: ")
print(" ")
print(" Tr_bath{", A, "W_eq}")
print(" ")
print("The expression is normalized by:")
print(" ")
print(" Tr_bath{", Anorm, "W_eq}")
<a|H(t)|b><c|H(t-tau)|d><d|W|e>
= <\Psi_e|<a|H(t)|b><c|H(t-tau)|d>|\Psi_d>
= <\Psi_g|Dagger(U_e(T))Dagger(U_a(t))U_b(t)Dagger(U_c(t-tau))U_d(t-tau)
x U_d(T)|\Psi_g>
= <\Psi_g|[U_g(T)Dagger(U_e(T))][Dagger(U_a(t))U_g(t)][Dagger(U_g(t)U_b(t)]
x [Dagger(U_c(t-tau))U_g(t-tau)][Dagger(U_g(t-tau)U_d(t-tau)]
x [U_d(T)Dagger(U_g(T))]|\Psi_g>
= Uged(e,T)*Uedg(a,t)*Ugde(b,t)*Uedg(c,t-tau)*Ugde(d,t-tau)*Uegd(d,T)
"""
A = Uged(e, T) * Uedg(a, t) * Ugde(b, t) * Uedg(c, t - tau) * Ugde(
d, t - tau) * Uegd(d, T)
Anorm = Uged(e, T) * Uegd(d, T)
verbatim = True
if verbatim:
print(" ")
print("Expression to evaluate: ")
print(" ")
print(" Tr_bath{", A, "W_eq}")
print(" ")
print("The expression is normalized by:")
print(" ")
print(" Tr_bath{", Anorm, "W_eq}")
print(" ")